8.7 Curved Mirrors

WARNING: OBJECTS IN MIRROR MAY BE CLOSER THAN THEY APPEAR.

If you’ve ever sat in the driver’s seat of a car, or indeed the one next to it, you are probably familiar with this sign, which is often printed on the rear-view mirrors. And if the car ride was long enough, perhaps you had time to wonder why this was the case. The driver, and any bored passengers, are subject to the powers of a curved mirror.

Curved mirrors are used by many of us everyday, from shaving and makeup mirrors, to the curved mirrors drivers use for seeing around a blind corner. Though they might not appear to be, they are in fact very similar to lenses. However, for the mirrors to follow these rules, they have to be evenly shaped, and therefore are usually made of the surface of a sphere shape, though they can also be parabolic. The mirror is given a name depending on whether it is the inside surface or the outside surface of the sphere that reflects.

Convex mirrors, opposite to lenses, are known as diverging mirrors, while concave mirrors cause parallel light rays to converge. The name of the mirror indicates the side of the sphere that is reflective.

In a concave lens, the point at which parallel light rays converge is once again called the focal point, sometimes known as the principal focus. From the law of reflection, where the angle of incidence must equal the angle of reflection, and the above idea, we know that:

1. A ray that is parallel to the principal axis will be reflected so that it passes through the focal point, and conversely,

2. A ray that passes through the focal point will be reflected so that it is parallel

to the principal axis

The focal point and the center of the would-be sphere of which the mirror was a part lie on a line that is also called the principal axis, as in lens diagrams.

In this diagram, the principal axis is pink, and the focal point is green. Notice that the focal point is not very close to the center of the sphere (shown here in two dimensions).

Ray diagrams can once again be drawn, to show how curved mirrors produce images. To construct the image of a point, only two rays are needed. If the focal point is known, we can construct two rays and their reflections without having to measure any angles. These are a ray that is parallel to the principal axis, and a ray that goes through the focal point.

In the figure above, a concave mirror is used to make a real, inverted image of the light purple arrow (symbolizing the object). Notice that in the case of mirrors, real images are located on the same side of the mirror as the object. A concave mirror can be used to make an enlarged or diminished real image of an object just as a convex lens can. This is because, after all, the rays in a mirror diagram can go both ways, and for real images, the object and the image can be interchanged.

In this second image, a concave mirror is used to make an upright, magnified, virtual image. Once again it acts in the same way as a convex lens. In this case, however, it was necessary to measure one angle, because drawing a ray through the focal point would not have been of much use. Also, because the reflected rays diverge, they are followed back (in red) to their apparent origin, the image of the tip of the purple arrow. This is another indication that the image produced is virtual. This concave mirror set up is often used in telescopes, and is the one you would use for shaving or doing your makeup.

The final ray diagram for curved mirrors shows a convex mirror creating an upright, diminished, virtual image, just as a concave lens can do. In fact, you might notice that this diagram is similar to the previous one, with only the image and the object switched. This diagram however, does not require that the focal point be known, though it does require two angles to be measured.

Because they are so similar to lenses, it is easy to expect that spherical mirrors also follow the lens equation. In fact, they do. However, it is important to keep in mind the differences between what is considered by convention to be real or virtual. A simple guideline for this is that if the item (e.g. focal point, image distance) in question is on the same side of the mirror as the object, it is considered to be real. Otherwise, it is virtual, and its value in the lens equation is said to be negative.

To get back to the original question, why are objects in the rear-view mirror closer than they appear? Because they appear to be smaller than they actually are. The kind of mirror used in the rear-view mirror must therefore be a convex lens, because it creates smaller, yet still upright images - after all, the cars you see are not upside down. This type of mirror is useful in a car because by making everything smaller, it allows the driver to see a greater range of things behind him.